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The present article offers a certain unifying approach to time series regression modelling by combining partial likelihood (PL) inference and generalized linear models. An advantage gained by resorting to PL is that the joint distribution of the response and the covariates is left unspecified, and furthermore, PL allows for temporal or sequential conditional inference with respect to a filtration generated by all that is known to the observer at the time of observation. Two real data examples illustrate the methodology.

Categorical---or qualitative---time series data with random time-dependent covariates are frequently encountered in diverse applications as the list of examples shows. As with "ordinary'' time series, the data analyst is faced with the same problems of modeling, estimation, model checking, diagnostics and prediction. The present work shows that these questions can be attacked by means of regression theory for categorical time series whose foundation is based on generalized linear models and partial likelihood inference. A variety of models are provided to illustrate the selection of the link function and recent large sample results are reviewed. The theory is developed without resorting to the Markov assumption and to the notion of stationarity. Moreover, regression methods for categorical time series allow for parsimonious modeling and incorporation of random time-dependent covariates as opposed to other procedures. In particular, nominal and ordinal time series are analyzed and compared empirically to Markov chains and mixture transition distribution models.

In this dissertation we propose a class of time series models for mixture data. We call these logistic mixtures. In such models the mixture's component densities have a generalized linear model (GLM) form. The regime probabilities are allowed to change over time and are modeled with a logistic regression structure. The regressors of both the component GLM distributions and the logistic probabilities may include covariates as well as past values of the process. We develop an EM algorithm for estimation, give conditions for consistency and asymptotic normality, examine the model through simulations, and apply it to rain rate data. Finally, we consider a likelihood ratio-based test for determining if the data arise from a logistic mixture versus the null hypothesis of the data coming from a single distribution (i.e. no mixture). Because the mixture probabilities are not constant, we are able to develop a test that avoids some of the problems associated with likelihood ratio tests of mixtures.

In this dissertation we propose a class of time series models for mixture data. We call these logistic mixtures. In such models the mixture's component densities have a generalized linear model (GLM) form. The regime probabilities are allowed to change over time and are modeled with a logistic regression structure. The regressors of both the component GLM distributions and the logistic probabilities may include covariates as well as past values of the process. We develop an EM algorithm for estimation, give conditions for consistency and asymptotic normality, examine the model through simulations, and apply it to rain rate data. Finally, we consider a likelihood ratio-based test for determining if the data arise from a logistic mixture versus the null hypothesis of the data coming from a single distribution (i.e. no mixture). Because the mixture probabilities are not constant, we are able to develop a test that avoids some of the problems associated with likelihood ratio tests of mixtures.

We present a generalized logistic regression model for the statistical analysis of multicategorical time series. The model is suitably parameterized and partial likelihood inference is proposed for estimation of the unknown parameters. A goodness of fit statistic is derived to judge the quality of fit. The analysis is applied to data from the Tropical Ocean and Global Atmosphere/Coupled Ocean‐Atmosphere Response Experiment.

In this dissertation we present extensions of Rice's formula for the expected zero-crossing rate of a Gaussian process to some useful non-Gaussian cases. In particular, we extend Rice's formula to the class of stationary processes which are a monotone transformation of a Gaussian process, to countable mixtures of Gaussians, and to products of independent Gaussian processes. In all the above mentioned cases the expected zero-crossing rates are given for both continuous time and discrete time processes. We also investigate the application of parametric filtering, using zero-crossing count statistics, to the problem of frequency estimation in a mixed spectrum model and the application of mean- level-crossing counts of the envelope of a Gaussian process to a radar detection problem. For the radar problem we prove asymptotic normality of the level-crossings of the envelope of a Gaussian process and provide and expression for the asymptotic variance.

We study regression models for nonstationary categorical time series and their applications, and address the issues of prediction, estimation and control. Generalized Linear Models and Partial Likelihood are the basic tools in the present study. The models link the probabilities of each category to a covariate process through a vector of time invariant parameters. Under mild regularity conditions, asymptotic properties of the estimators are established by appealing to martingale theory, and certain diagnostic tools are presented for checking the model adequacy. The methodology is demonstrated using real rainfall data. Subsequently we discuss a new recursive estimation method for time series following generalized linear models, motivated by the logistic regression model in conjunction with binary time series. The estimation procedure, suitably modified, gives rise to a stochastic approximation scheme used here to illustrate a connection between control theory and generalized linear models.

In this dissertation we present extensions of Rice's formula for the expected zero-crossing rate of a Gaussian process to some useful non-Gaussian cases. In particular, we extend Rice's formula to the class of stationary processes which are a monotone transformation of a Gaussian process, to countable mixtures of Gaussians, and to products of independent Gaussian processes. In all the above mentioned cases the expected zero-crossing rates are given for both continuous time and discrete time processes. We also investigate the application of parametric filtering, using zero-crossing count statistics, to the problem of frequency estimation in a mixed spectrum model and the application of mean- level-crossing counts of the envelope of a Gaussian process to a radar detection problem. For the radar problem we prove asymptotic normality of the level-crossings of the envelope of a Gaussian process and provide and expression for the asymptotic variance.

We study regression models for nonstationary categorical time series and their applications, and address the issues of prediction, estimation and control. Generalized Linear Models and Partial Likelihood are the basic tools in the present study. The models link the probabilities of each category to a covariate process through a vector of time invariant parameters. Under mild regularity conditions, asymptotic properties of the estimators are established by appealing to martingale theory, and certain diagnostic tools are presented for checking the model adequacy. The methodology is demonstrated using real rainfall data. Subsequently we discuss a new recursive estimation method for time series following generalized linear models, motivated by the logistic regression model in conjunction with binary time series. The estimation procedure, suitably modified, gives rise to a stochastic approximation scheme used here to illustrate a connection between control theory and generalized linear models.

A general parametric filtering procedure (the PF method) is proposed for the problem of multiple frequency estimation in mixed-spectrum times series (i.e., superimposed sinusoids in additive noise). The method is based on the fact that a sum of sinusoids satisfies an homogeneous autoregressive (AR) equation. The gist of the method is to parametrize a linear filter so that it possesses a certain parametrization property as suggested by the particular form of the bias encountered by Prony's (least squares) estimator. For any parametric filter with this property, in addition to some mild regularity conditions, the least squares estimator from the filtered data, as a function of the filter parameter, constitutes a contractive mapping - whose multivariate fixed-point serves as a consistent AR estimator. The chronic bias of Prony's estimator is thus eliminated. Coupled with the all- pole (AR) filter endowed with an extra bandwidth parameter, the PF method can achieve the accuracy of nonlinear least squares by a simple iterative procedure consisting of linear least squares estimation followed by linear recursive filtering. Crude initial guesses such as those from Prony's estimator are sufficient to initiate the iteration. The method is also capable of resolving closely-spaced frequencies which are unresolvable by periodogram analysis or DFT.To analyze the statistical properties of the PF method, some classical asymptotic results concerning the sample autocovariances are extended to accommodate mixed-spectrum time series and parametric filtering. In particular, under regularity conditions, uniform strong consistency and asymptotic normality are proved for the sample autocovariances of a mixed- spectrum time series after parametric filtering. Equipped with these results, some statistical properties of the PF method itself are investigated. These include the existence of the PF estimator as a fixed-point of the parametric least squares mapping, the convergence of an iterat

A general parametric filtering procedure (the PF method) is proposed for the problem of multiple frequency estimation in mixed-spectrum times series (i.e., superimposed sinusoids in additive noise). The method is based on the fact that a sum of sinusoids satisfies an homogeneous autoregressive (AR) equation. The gist of the method is to parametrize a linear filter so that it possesses a certain parametrization property as suggested by the particular form of the bias encountered by Prony's (least squares) estimator. For any parametric filter with this property, in addition to some mild regularity conditions, the least squares estimator from the filtered data, as a function of the filter parameter, constitutes a contractive mapping - whose multivariate fixed-point serves as a consistent AR estimator. The chronic bias of Prony's estimator is thus eliminated. Coupled with the all- pole (AR) filter endowed with an extra bandwidth parameter, the PF method can achieve the accuracy of nonlinear least squares by a simple iterative procedure consisting of linear least squares estimation followed by linear recursive filtering. Crude initial guesses such as those from Prony's estimator are sufficient to initiate the iteration. The method is also capable of resolving closely-spaced frequencies which are unresolvable by periodogram analysis or DFT.To analyze the statistical properties of the PF method, some classical asymptotic results concerning the sample autocovariances are extended to accommodate mixed-spectrum time series and parametric filtering. In particular, under regularity conditions, uniform strong consistency and asymptotic normality are proved for the sample autocovariances of a mixed- spectrum time series after parametric filtering. Equipped with these results, some statistical properties of the PF method itself are investigated. These include the existence of the PF estimator as a fixed-point of the parametric least squares mapping, the convergence of an iterat

This research dealt with higher order crossings, quantities which were defined and proved to be useful in discrimination in time series. In particular the Higher Order Crossings Theorem has been proved and a new goodness of fit and discrimination statistic has been suggested and applied in testing model adequacy in ARIMA processes, and in discrimination in EEG data. A connection with an application to particles arrangements in physics has been found and a quantity called an m'th order unit has been defined. This has been applied in finding the distribution of rare events in Binary Series.

The purpose of this document is to introduce a graphical device useful as a measure of similarity or as a goodness of fit criterion for hypothesized time series models. It is based on the actual oscillation observed in time series as depicted by axis-crossings and higher order crossings. Higher order crossings (HOC) are axis-crossings of differenced time series and are closely linked to the spectral content of the series. In fact under the Gaussian assumption, to which we shall adhere, HOC determine the finite dimensional distributions up to a scale parameter given that the mean is zero. The main advantage of HOC is that they are easily obtained from an observed series and that only very few of them are needed, as the dicriminatory power in HOC usually diminishes with their order. Higher order crossings in time series discrimination were discussed in Kedem and Slud (1981), (1982), where a certain goodness of fit criterion is suggested, Here however the emphasis is on a graphical device rather than a single test statistic. This graphical method may be shown useful in answering the question 'Does a given time series oscillate as a certain hypothesized model?' Some examples with real and simulated data demonstrate the use and potential of this method.

A graphical device that is useful as a diagnostic tool for higher order crossings was introduced. The graphical tool is also useful as a descriptive similarity measure for time series models. Higher order crossings analysis has been applied to real data, revealing several known periodicities as well as several previously undiscovered periodicities. The class of strictly oscillatory processes was introduced.

This research focused on higher order crossings of time series and their application in spectral analysis, and on the various point processes obtained from stationary processes and their relations to reliability models. The higher order crossings theorem was extended to two dimensions, and the one-dimensional version was used to determine the connection between axis crossings and the frequency content of a stationary process. Considerable progress was made in calculating the asymptotic variance-covariance matrix for the vector of higher-order crossing counts, and time series analysis of highly nonlinear and chaotic stationary processes was carried out. The report summarizes progress in these areas. (Author)